(x-1)^3-(x+2)(x^2-2x+4)+3(x+4)(x-4)

3 min read Jun 17, 2024

Simplifying the Expression: (x-1)^3 - (x+2)(x^2 - 2x + 4) + 3(x+4)(x-4)

This article will guide you through simplifying the given algebraic expression: (x-1)^3 - (x+2)(x^2 - 2x + 4) + 3(x+4)(x-4).

Step 1: Expand the Cubed Term

We begin by expanding the cubed term, (x-1)^3, using the binomial theorem or by repeated multiplication:

(x-1)^3 = (x-1)(x-1)(x-1) = (x^2 - 2x + 1)(x-1) = x^3 - 3x^2 + 3x - 1

Step 2: Expand the Products Using the Difference of Cubes and Difference of Squares Formulas

The expression (x+2)(x^2 - 2x + 4) represents the sum of cubes, and (x+4)(x-4) represents the difference of squares. We can utilize the following formulas:

Sum of Cubes: a^3 + b^3 = (a+b)(a^2 - ab + b^2)
Difference of Squares: a^2 - b^2 = (a+b)(a-b)

Applying these formulas, we get:

(x+2)(x^2 - 2x + 4) = x^3 + 8
3(x+4)(x-4) = 3(x^2 - 16) = 3x^2 - 48

Step 3: Substitute and Combine Like Terms

Now, substitute the expanded terms back into the original expression:

(x-1)^3 - (x+2)(x^2 - 2x + 4) + 3(x+4)(x-4) = (x^3 - 3x^2 + 3x - 1) - (x^3 + 8) + (3x^2 - 48)

Combine the like terms:

= x^3 - 3x^2 + 3x - 1 - x^3 - 8 + 3x^2 - 48

= 3x - 57

Conclusion

Therefore, the simplified form of the expression (x-1)^3 - (x+2)(x^2 - 2x + 4) + 3(x+4)(x-4) is 3x - 57.